1) Notice that the result in Theorem REF  is also valid as \(\alpha =0\) , and even when the hypothesis A1) is replaced with the more general condition \(\tau _m(k) = 1 + k^{-1}\alpha +o(k^{-1})\) , as \(\ k \rightarrow \infty \) , \(\alpha \ge 0\) , being the calculation in this latter scenario a little more cumbersome. In the case \(\alpha =0\)  and \(m=1\)  the result provides an alternative proof of the weak convergence result for the BGW process (see [1]}, p. 388) for a non-array version.
